# How do you evaluate 4log_4 6 - log_4 5?

May 29, 2015

Use the following properties of logs:
1] $a \log x = \log {x}^{a}$
2] $\log x - \log y = \log \left(\frac{x}{y}\right)$
So in your case:
${\log}_{4} \left({6}^{4}\right) - {\log}_{4} \left(5\right) = {\log}_{4} \left({6}^{4} / 5\right) = {\log}_{4} \left(259.2\right) = x$
so:
${4}^{x} = 259.2$
if $x = 4$
${4}^{4} = 256$
if $x = 4.002$
${4}^{4.002} = 259.2$

Another thing that you can do is to change base (using, for example, natural logs) as:
${\log}_{4} \left(259.2\right) = \ln \frac{259.2}{\ln \left(4\right)} = 4.0089$